The study of isometric actions of groups has proved to be a fundamental tool in representation theory as in geometric group theory. A locally compact group has the Haagerup Property if it possesses an isometric action on a Hilbert space, which is metrically proper. At the opposite, it has the Kazhdan Property if every such action has a fixed point. This work is, in a large part, devoted to interpolate between these two properties, by introducing the relative Kazhdan Property for any pair (G,X), X being any subset of the locally compact group G.

This generalizes Margulis' relative Kazhdan Property for pairs (G,H) where H is a subgroup of G. However, we give the first known example of a group G having unbounded subsets with the relative Kazhdan Property (and therefore failing to have the Haagerup Property), but none of whose can be chosen as a subgroup. A thorough study of this relative Property is carried out for Lie groups and their lattices.